One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of k disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall’s conjecture on packing disjoint dijoins in a directed graph, or Rota’s beautiful conjecture on rearrangements of bases. In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into k common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when $$k=2$$ k = 2 . Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when $$k=2$$ k = 2 , one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.

中文翻译：

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在拟阵中打包公共碱基的复杂性

拟阵优化中最有趣的未解决问题之一是表征两个拟阵的 k 个不相交公共基的存在。一长串可以表述为特殊情况的猜想很好地说明了这个问题的重要性，例如 Woodall 的关于在有向图中打包不相交双连接的猜想，或 Rota 关于基重排的美丽猜想。在本文中，我们证明了这个问题在秩预言模型下是困难的，即我们证明没有算法可以决定是否可以使用多项式多项式将两个拟阵的公共基础集划分为 k 个公共基查询。我们的复杂性结果甚至适用于 $$k=2$$k = 2 的非常特殊的情况。通过一系列的削减，我们还展示了在两个拟阵中打包公共基的抽象问题包括 NAE-SAT 问题和有向图中的完美偶数因子问题。这些结果反过来意味着问题不仅在独立预言模型中很困难，而且当 $$k=2$$k = 2 时已经包括 NP-完全特殊情况，其中一个拟阵是分区拟阵，而另一个拟阵是线性的，并由显式表示给出。